algebraic surfaces and hyperbolic geometry

Current Developments in Algebraic GeometryMSRI PublicationsVolume 59, 2011

Algebraic surfaces and hyperbolic geometryBURT TOTARO

We describe the Kawamata–Morrison cone conjecture on the structure ofCalabi–Yau varieties and more generally klt Calabi–Yau pairs. The conjectureis true in dimension 2. We also show that the automorphism group of a K3surface need not be commensurable with an arithmetic group, which answersa question by Mazur.

1. Introduction

Many properties of a projective algebraic variety can be encoded by convexcones, such as the ample cone and the cone of curves. This is especially usefulwhen these cones have only finitely many edges, as happens for Fano varieties.For a broader class of varieties which includes Calabi–Yau varieties and manyrationally connected varieties, the Kawamata–Morrison cone conjecture predictsthe structure of these cones. I like to think of this conjecture as what comesafter the abundance conjecture. Roughly speaking, the cone theorem of Mori,Kawamata, Shokurov, Kollár, and Reid describes the structure of the curves on aprojective variety X on which the canonical bundle K X has negative degree; theabundance conjecture would give strong information about the curves on whichK X has degree zero; and the cone conjecture fully describes the structure of thecurves on which K X has degree zero.

We give a gentle summary of the proof of the cone conjecture for algebraicsurfaces, with plenty of examples [Totaro 2010]. For algebraic surfaces, thesecones are naturally described using hyperbolic geometry, and the proof can alsobe formulated in those terms.

Example 7.3 shows that the automorphism group of a K3 surface need notbe commensurable with an arithmetic group. This answers a question by BarryMazur [1993, Section 7].

Thanks to John Christian Ottem, Artie Prendergast-Smith, and Marcus Zi-browius for their comments.

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2. The main trichotomy

Let X be a smooth complex projective variety. There are three main types ofvarieties. (Not every variety is of one of these three types, but minimal modeltheory relates every variety to one of these extreme types.)

Fano. This means that −K X is ample. (We recall the definition of amplenessin Section 3.)

Calabi–Yau. We define this to mean that K X is numerically trivial.ample canonical bundle. This means that K X is ample; it implies that X is of

“general type.”Here, for X of complex dimension n, the canonical bundle K X is the line bun-

dle �nX of n-forms. We write −K X for the dual line bundle K ∗X , the determinant

of the tangent bundle.

Example 2.1. Let X be a curve, meaning that X has complex dimension 1.Then X is Fano if it has genus zero, or equivalently if X is isomorphic to thecomplex projective line P1; as a topological space, this is the 2-sphere. Next, Xis Calabi–Yau if X is an elliptic curve, meaning that X has genus 1. Finally, Xhas ample canonical bundle if it has genus at least 2.

Example 2.2. Let X be a smooth surface in P3. Then X is Fano if it has degreeat most 3. Next, X is Calabi–Yau if it has degree 4; this is one class of K3surfaces. Finally, X has ample canonical bundle if it has degree at least 5.

Belonging to one of these three classes of varieties is equivalent to the existenceof a Kähler metric with Ricci curvature of a given sign [Yau 1978]. Precisely,a smooth projective variety is Fano if and only if it has a Kähler metric withpositive Ricci curvature; it is Calabi–Yau if and only if it has a Ricci-flat Kählermetric; and it has ample canonical bundle if and only if it has a Kähler metricwith negative Ricci curvature.

We think of Fano varieties as the most special class of varieties, with projectivespace as a basic example. Strong support for this idea is provided by Kollár,Miyaoka, and Mori’s theorem that smooth Fano varieties of dimension n form abounded family [Kollár et al. 1992]. In particular, there are only finitely manydiffeomorphism types of smooth Fano varieties of a given dimension.

Example 2.3. Every smooth Fano surface is isomorphic to P1× P1 or to a

blow-up of P2 at at most 8 points. The classification of smooth Fano 3-folds isalso known, by Iskovskikh, Mori, and Mukai; there are 104 deformation classes[Iskovkikh and Prokhorov 1999].

By contrast, varieties with ample canonical bundle form a vast and uncontrol-lable class. Even in dimension 1, there are infinitely many topological types ofvarieties with ample canonical bundle (curves of genus at least 2). Calabi–Yau

ALGEBRAIC SURFACES AND HYPERBOLIC GEOMETRY 407

varieties are on the border in terms of complexity. It is a notorious open questionwhether there are only finitely many topological types of smooth Calabi–Yauvarieties of a given dimension. This is true in dimension at most 2. In particular,a smooth Calabi–Yau surface is either an abelian surface, a K3 surface, or aquotient of one of these surfaces by a free action of a finite group (and onlyfinitely many finite groups occur this way).

3. Ample line bundles and the cone theorem

After a quick review of ample line bundles, this section states the cone theoremand its application to Fano varieties. Lazarsfeld’s book [2004] is an excellentreference on ample line bundles.

Definition 3.1. A line bundle L on a projective variety X is ample if somepositive multiple nL (meaning the line bundle L⊗n) has enough global sectionsto give a projective embedding

X ↪→ PN .

(Here N = dimC H 0(X, nL)− 1.)

One reason to investigate which line bundles are ample is in order to classifyalgebraic varieties. For classification, it is essential to know how to describe avariety with given properties as a subvariety of a certain projective space definedby equations of certain degrees.

Example 3.2. For X a curve, L is ample on X if and only if it has positivedegree. We write L · X = deg(L|X ) ∈ Z.

An R-divisor on a smooth projective variety X is a finite sum

D =∑

ai Di

with ai ∈ R and each Di an irreducible divisor (codimension-one subvariety) inX . Write N 1(X) for the “Néron–Severi” real vector space of R-divisors modulonumerical equivalence: D1 ≡ D2 if D1 ·C = D2 ·C for all curves C in X . (Forme, a curve is irreducible.)

We can also define N 1(X) as the subspace of the cohomology H 2(X,R)

spanned by divisors. In particular, it is a finite-dimensional real vector space.The dual vector space N1(X) is the space of 1-cycles

∑ai Ci modulo numerical

equivalence, where Ci are curves on X . We can identify N1(X) with the subspaceof the homology H2(X,R) spanned by algebraic curves.

Definition 3.3. The closed cone of curves Curv(X) is the closed convex cone inN1(X) spanned by curves on X .

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Definition 3.4. An R-divisor D is nef if D ·C ≥0 for all curves C in X . Likewise,a line bundle L on X is nef if the class [L] of L (also called the first Chern classc1(L)) in N 1(X) is nef. That is, L has nonnegative degree on all curves in X .

Thus Nef(X)⊂ N 1(X) is a closed convex cone, the dual cone to Curv(X)⊂N1(X).

Theorem 3.5 (Kleiman). A line bundle L is ample if and only if [L] is in theinterior of the nef cone in N 1(X).

This is a numerical characterization of ampleness. It shows that we know theample cone Amp(X)⊂ N 1(X) if we know the cone of curves Curv(X)⊂ N1(X).The following theorem gives a good understanding of the “K -negative” half ofthe cone of curves [Kollár and Mori 1998, Theorem 3.7]. A rational curve meansa curve that is birational to P1.

Theorem 3.6. (Cone theorem: Mori, Shokurov, Kawamata, Reid, Kollár). LetX be a smooth projective variety. Write K<0

X = {u ∈ N1(X) : K X · u < 0}. Thenevery extremal ray of Curv(X)∩ K<0

X is isolated, spanned by a rational curve,and can be contracted.

In particular, every extremal ray of Curv(X) ∩ K<0X is rational (meaning

that it is spanned by a Q-linear combination of curves, not just an R-linearcombination), since it is spanned by a single curve. A contraction of a normalprojective variety X means a surjection from X onto a normal projective variety Ywith connected fibers. A contraction is determined by a face of the cone of curvesCurv(X), the set of elements of Curv(X) whose image under the pushforwardmap N1(X)→ N1(Y ) is zero. The last statement in the cone theorem means thatevery extremal ray in the K -negative half-space corresponds to a contraction ofX .

Corollary 3.7. For a Fano variety X , the cone of curves Curv(X) (and thereforethe dual cone Nef(X)) is rational polyhedral.

A rational polyhedral cone means the closed convex cone spanned by finitelymany rational points.

Proof. Since −K X is ample, K X is negative on all of Curv(X)− {0}. So thecone theorem applies to all the extremal rays of Curv(X). Since they are isolatedand live in a compact space (the unit sphere), Curv(X) has only finitely manyextremal rays. The cone theorem also gives that these rays are rational. �

It follows, in particular, that a Fano variety has only finitely many differentcontractions. A simple example is the blow-up X of P2 at one point, whichis Fano. In this case, Curv(X) is a closed strongly convex cone in the two-dimensional real vector space N1(X), and so it has exactly two 1-dimensional


faces. We can write down two contractions of X , X → P2 (contracting a(−1)-curve) and X → P1 (expressing X as a P1-bundle over P1). Each ofthese morphisms must contract one of the two 1-dimensional faces of Curv(X).Because the cone has no other nontrivial faces, these are the only nontrivialcontractions of X .

4. Beyond Fano varieties

“Just beyond” Fano varieties, the cone of curves and the nef cone need not berational polyhedral. Lazarsfeld’s book [2004] gives many examples of this type,as do other books on minimal model theory [Debarre 2001; Kollár and Mori1998].

Example 4.1. Let X be the blow-up of P2 at n very general points. For n ≤ 8,X is Fano, and so Curv(X) is rational polyhedral. In more detail, for 2≤ n ≤ 8,Curv(X) is the convex cone spanned by the finitely many (−1)-curves in X .(A (−1)-curve on a surface X means a curve C isomorphic to P1 with self-intersection number C2

=−1.) For example, when n = 6, X can be identifiedwith a cubic surface, and the (−1)-curves are the famous 27 lines on X .

But for n≥9, X is not Fano, since (−K X )2=9−n (whereas a projective variety

has positive degree with respect to any ample line bundle). For p1, . . . , pn verygeneral points in P2, X contains infinitely many (−1)-curves; see [Hartshorne1977, Exercise V.4.15]. Every curve C with C2< 0 on a surface spans an isolatedextremal ray of Curv(X), and so Curv(X) is not rational polyhedral.

Notice that a (−1)-curve C has K X ·C = −1, and so these infinitely manyisolated extremal rays are on the “good” (K -negative) side of the cone of curves,in the sense of the cone theorem. The K -positive side is a mystery. It isconjectured (Harbourne–Hirschowitz) that the closed cone of curves of a verygeneral blow-up of P2 at n ≥ 10 points is the closed convex cone spanned by the(−1)-curves and the “round” positive cone {x ∈ N1(X) : x2

≥ 0 and H · x ≥ 0},where H is a fixed ample line bundle. This includes the famous Nagata conjecture[Lazarsfeld 2004, Remark 5.1.14] as a special case. By de Fernex, even if theHarbourne–Hirschowitz conjecture is correct, the intersection of Curv(X) withthe K -positive half-space, for X a very general blow-up of P2 at n ≥ 11 points,is bigger than the intersection of the positive cone with the K -positive half-space,because the (−1)-curves stick out a lot from the positive cone [de Fernex 2010].

Example 4.2. Calabi–Yau varieties (varieties with K X ≡0) are also “just beyond”Fano varieties (−K X ample). Again, the cone of curves of a Calabi–Yau varietyneed not be rational polyhedral.

For example, let X be an abelian surface, so X ∼= C2/3 for some lattice3∼= Z4 such that X is projective. Then Curv(X)= Nef(X) is a round cone, the

410 BURT TOTARO

positive cone

{x ∈ N 1(X) : x2≥ 0 and H · x ≥ 0},

where H is a fixed ample line bundle. (Divisors and 1-cycles are the same thingon a surface, and so the cones Curv(X) and Nef(X) lie in the same vector spaceN 1(X).) Thus the nef cone is not rational polyhedral if X has Picard numberρ(X) := dimR N 1(X) at least 3 (and sometimes when ρ = 2).

For a K3 surface, the closed cone of curves may be round, or may be theclosed cone spanned by the (−2)-curves in X . (One of those two properties musthold, by [Kovács 1994].) There may be finitely or infinitely many (−2)-curves.See Section 5.1 for an example.

5. The cone conjecture

But there is a good substitute for the cone theorem for Calabi–Yau varieties, theMorrison–Kawamata cone conjecture. In dimension 2, this is a theorem of Sterk,Looijenga, and Namikawa [Sterk 1985; Namikawa 1985; Kawamata 1997]. Wecall this Sterk’s theorem for convenience:

Theorem 5.1. Let X be a smooth complex projective Calabi–Yau surface (mean-ing that K X is numerically trivial). Then the action of the automorphism groupAut(X) on the nef cone Nef(X)⊂ N 1(X) has a rational polyhedral fundamentaldomain.

Remark 5.2. For any variety X , if Nef(X) is rational polyhedral, then the groupAut∗(X) := im(Aut(X)→GL(N 1(X))) is finite. This is easy: the group Aut∗(X)must permute the set consisting of the smallest integral point on each extremal rayof Nef(X). Sterk’s theorem implies the remarkable statement that the converseis also true for Calabi–Yau surfaces. That is, if the cone Nef(X) is not rationalpolyhedral, then Aut∗(X) must be infinite. Note that Aut∗(X) coincides withthe discrete part of the automorphism group of X up to finite groups, becauseker(Aut(X)→ GL(N 1(X))) is an algebraic group and hence has only finitelymany connected components.

Sterk’s theorem should generalize to Calabi–Yau varieties of any dimension(the Morrison–Kawamata cone conjecture). But in dimension 2, we can visualizeit better, using hyperbolic geometry.

Indeed, let X be any smooth projective surface. The intersection form onN 1(X) always has signature (1, n) for some n (the Hodge index theorem). So{x ∈ N 1(X) : x2 > 0} has two connected components, and the positive cone{x ∈ N 1(X) : x2 > 0 and H · x > 0} is the standard round cone. As a result, wecan identify the quotient of the positive cone by R>0 with hyperbolic n-space.


One way to see this is that the negative of the Lorentzian metric on N 1(X)=R1,n

restricted to the quadric {x2= 1} is a Riemannian metric with curvature −1.

For any projective surface X , Aut(X) preserves the intersection form onN 1(X). So Aut∗(X) is always a group of isometries of hyperbolic n-space,where n = ρ(X)− 1.

By definition, two groups G1 and G2 are commensurable, written G1.= G2,

if some finite-index subgroup of G1 is isomorphic to a finite-index subgroup ofG2. A group virtually has a given property if some subgroup of finite index hasthe property. Since the groups we consider are all virtually torsion-free, we arefree to replace a group G by G/N for a finite normal subgroup N (that is, Gand G/N are commensurable).

5.1. Examples. For an abelian surface X with Picard number at least 3, thecone Nef(X) is round, and so Aut∗(X) must be infinite by Sterk’s theorem. (Forabelian surfaces, the possible automorphism groups were known long before;see [Mumford 1970, Section 21].)

For example, let X = E× E with E an ellip-tic curve (not having complex multiplication).Then ρ(X) = 3, with N 1(X) spanned by thecurves E × 0, 0× E , and the diagonal 1E inE × E . So Aut∗(X) must be infinite. In fact,

Aut∗(X)∼= PGL(2,Z).

Here GL(2,Z) acts on E × E as on the direct sum of any abelian group withitself. This agrees with Sterk’s theorem, which says that Aut∗(X) acts on thehyperbolic plane with a rational polyhedral fundamental domain; a fundamentaldomain for PGL(2,Z) acting on the hyperbolic plane (not preserving orientation)is given by any of the triangles in the figure.

For a K3 surface, the cone Nef(X) may or may not be the whole positive cone.For any projective surface, the nef cone modulo scalars is a convex subset ofhyperbolic space. A finite polytope in hyperbolic space (even if some vertices areat infinity) has finite volume. So Sterk’s theorem implies that, for a Calabi–Yausurface, Aut∗(X) acts with finite covolume on the convex set Nef(X)/R>0 inhyperbolic space.

P

For example, let X be a K3 surface such that Pic(X)is isomorphic to Z3 with intersection form0 1 1

1 −2 01 0 −2

.

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Such a surface exists, since Nikulin showed that every even lattice of rank atmost 10 with signature (1, ∗) is the Picard lattice of some complex projective K3surface [Nikulin 1979, Section 1.12]. Using the ideas of Section 6, one computesthat the nef cone of X modulo scalars is the convex subset of the hyperbolicplane shown in the figure. The surface X has a unique elliptic fibration X→ P1,given by a nef line bundle P with 〈P, P〉 = 0. The line bundle P appears inthe figure as the point where Nef(X)/R>0 meets the circle at infinity. And Xcontains infinitely many (−2)-curves, whose orthogonal complements are thecodimension-1 faces of the nef cone. Sterk’s theorem says that Aut(X) must acton the nef cone with rational polyhedral fundamental domain. In this example,one computes that Aut(X) is commensurable with the Mordell–Weil group ofthe elliptic fibration (Pic0 of the generic fiber of X→ P1), which is isomorphicto Z. One also finds that all the (−2)-curves in X are sections of the ellipticfibration. The Mordell–Weil group moves one section to any other section, andso it divides the nef cone into rational polyhedral cones as in the figure.

6. Outline of the proof of Sterk’s theorem

We discuss the proof of Sterk’s theorem for K3 surfaces. The proof for abeliansurfaces is the same, but simpler (since an abelian surface contains no (−2)-curves), and these cases imply the case of quotients of K3 surfaces or abeliansurfaces by a finite group. For details, see [Kawamata 1997], based on the earlier[Sterk 1985; Namikawa 1985].

The proof of Sterk’s theorem for K3 surfaces relies on the Torelli theorem ofPiatetski-Shapiro and Shafarevich. That is, any isomorphism of Hodge structuresbetween two K3s is realized by an isomorphism of K3s if it maps the nef coneinto the nef cone. In particular, this lets us construct automorphisms of a K3surface X : up to finite index, every element of the integral orthogonal groupO(Pic(X)) that preserves the cone Nef(X) is realized by an automorphism ofX . (Here Pic(X)∼= Zρ , and the intersection form has signature (1, ρ(X)− 1) onPic(X).)

Moreover, Nef(X)/R>0 is a very special convex set in hyperbolic space Hρ−1:it is the closure of a Weyl chamber for a discrete reflection group W acting onHρ−1. We can define W as the group generated by all reflections in vectorsx ∈ Pic(X) with x2

=−2, or (what turns out to be the same) the group generatedby reflections in all (−2)-curves in X . By the first description, W is a normalsubgroup of O(Pic(X)). In fact, up to finite groups, O(Pic(X)) is the semidirectproduct group

O(Pic(X)) .= Aut(X)n W.


By general results on arithmetic groups going back to Minkowski, O(Pic(X))acts on the positive cone in N 1(X) with a rational polyhedral fundamentaldomain D. (This fundamental domain is not at all unique.) And the reflectiongroup W acts on the positive cone with fundamental domain the nef cone of X .Therefore, after we arrange for D to be contained in the nef cone, Aut(X) mustact on the nef cone with the same rational polyhedral fundamental domain D,up to finite index. Sterk’s theorem is proved.

7. Nonarithmetic automorphism groups

In this section, we show for the first time that the discrete part of the automor-phism group of a smooth projective variety need not be commensurable with anarithmetic group. (Section 5 defines commensurability.) This answers a questionraised by Mazur [1993, Section 7]. Corollary 7.2 applies to a large class of K3surfaces.

An arithmetic group is a subgroup of the group of Q-points of some Q-algebraic group HQ which is commensurable with H(Z) for some integralstructure on HQ; this condition is independent of the integral structure [Serre1979]. We view arithmetic groups as abstract groups, not as subgroups of a fixedLie group.

Borcherds gave an example of a K3 surface whose automorphism group isnot isomorphic to an arithmetic group [Borcherds 1998, Example 5.8]. But, ashe says, the automorphism group in his example has a nonabelian free subgroupof finite index, and so it is commensurable with the arithmetic group SL(2,Z).Examples of K3 surfaces with explicit generators of the automorphism grouphave been given by Keum, Kondo, Vinberg, and others; see [Dolgachev 2008,Section 5] for a survey.

Although they need not be commensurable with arithmetic groups, the auto-morphism groups G of K3 surfaces are very well-behaved in terms of geometricgroup theory. More generally this is true for the discrete part G of the automor-phism group of a surface X which can be given the structure of a klt Calabi–Yaupair, as defined in Section 8. Namely, G acts cocompactly on a CAT(0) space(a precise notion of a metric space with nonpositive curvature). Indeed, the nefcone modulo scalars is a closed convex subset of hyperbolic space, and thusa CAT(−1) space [Bridson and Haefliger 1999, Example II.1.15]. Removinga G-invariant set of disjoint open horoballs gives a CAT(0) space on which Gacts properly and cocompactly, by the proof of [Bridson and Haefliger 1999,Theorem II.11.27]. This implies all the finiteness properties one could want,even though G need not be arithmetic. In particular: G is finitely presented, afinite-index subgroup of G has a finite CW complex as classifying space, and

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G has only finitely many conjugacy classes of finite subgroups [Bridson andHaefliger 1999, Theorem III.0.1.1].

For smooth projective varieties in general, very little is known. For example,is the discrete part G of the automorphism group always finitely generated? Thequestion is open even for smooth projective rational surfaces. About the onlything one can say for an arbitrary smooth projective variety X is that G moduloa finite group injects into GL(ρ(X),Z), by the comments in Section 5.

In Theorem 7.1, a lattice means a finitely generated free abelian group with asymmetric bilinear form that is nondegenerate ⊗Q.

Theorem 7.1. Let M be a lattice of signature (1, n) for n ≥ 3. Let G be asubgroup of infinite index in O(M). Suppose that G contains Zn−1 as a subgroupof infinite index. Then G is not commensurable with an arithmetic group.

Corollary 7.2. Let X be a K3 surface over any field, with Picard number atleast 4. Suppose that X has an elliptic fibration with no reducible fibers and asecond elliptic fibration with Mordell–Weil rank positive. (For example, the latterproperty holds if the second fibration also has no reducible fibers.) Suppose alsothat X contains a (−2)-curve. Then the automorphism group of X is a discretegroup that is not commensurable with an arithmetic group.

Example 7.3. Let X be the double cover of P1×P1= {([x, y], [u, v])} ramified

along the following curve of degree (4, 4):

0= 16x4u4+ xy3u4

+ y4u3v− 40x4u2v2− x3 yu2v2

− x2 y2uv3

+ 33x4v4− 10x2 y2v4

+ y4v4.

Then X is a K3 surface whose automorphism group (over Q, or over Q) is notcommensurable with an arithmetic group.

Proof of Theorem 7.1. We can view O(M) as a discrete group of isometriesof hyperbolic n-space. Every solvable subgroup of O(M) is virtually abelian[Bridson and Haefliger 1999, Corollary II.11.28 and Theorem III.0.1.1]. By theclassification of isometries of hyperbolic space as elliptic, parabolic, or hyperbolic[Alekseevskij et al. 1993], the centralizer of any subgroup Z⊂ O(M) is eithercommensurable with Z (if a generator g of Z is hyperbolic) or commensurablewith Za for some a ≤ n − 1 (if g is parabolic). These properties pass to thesubgroup G of O(M). Also, G is not virtually abelian, because it contains Zn−1

as a subgroup of infinite index, and Zn−1 is the largest abelian subgroup of O(M)up to finite index. Finally, G acts properly and not cocompactly on hyperbolicn-space, and so G has virtual cohomological dimension at most n− 1 [Brown1982, Proposition VIII.8.1].

Suppose that G is commensurable with some arithmetic group 0. Thus 0 isa subgroup of the group of Q-points of some Q-algebraic group HQ, and 0 is


commensurable with H(Z) for some integral structure on HQ. We freely change0 by finite groups in what follows. So we can assume that HQ is connected. Afterreplacing HQ by the kernel of some homomorphism to a product of copies ofthe multiplicative group Gm over Q, we can assume that 0 is a lattice in the realgroup H(R) (meaning that vol(H(R)/0) <∞), by Borel and Harish-Chandra[Borel and Harish-Chandra 1962, Theorem 9.4].

Every connected Q-algebraic group HQ is a semidirect product RQ n UQ

where RQ is reductive and UQ is unipotent [Borel and Serre 1964, Theorem5.1]. By the independence of the choice of integral structure, we can assumethat 0 = R(Z)n U (Z) for some arithmetic subgroups R(Z) of RQ and U (Z) ofUQ. Since every solvable subgroup of G is virtually abelian, UQ is abelian, andU (Z) ∼= Za for some a. The conjugation action of RQ on UQ must be trivial;otherwise 0 would contain a solvable group of the form Z n Za which is notvirtually abelian. Thus 0 = R(Z)×Za . But the properties of centralizers in Gimply that any product group of the form W×Z contained in G must be virtuallyabelian. Therefore, a = 0 and HQ is reductive.

Modulo finite groups, the reductive group HQ is a product of Q-simple groupsand tori, and 0 is a corresponding product modulo finite groups. Since anyproduct group of the form W×Z contained in G is virtually abelian, HQ must beQ-simple. Since the lattice 0 in H(R) is isomorphic to the discrete subgroup Gof O(M)⊂ O(n, 1) (after passing to finite-index subgroups), Prasad showed thatdim(H(R)/K H ) ≤ dim(O(n, 1)/O(n))= n, where K H is a maximal compactsubgroup of H(R). Moreover, since G has infinite index in O(M) and henceinfinite covolume in O(n, 1), Prasad showed that either dim(H(R)/K H )≤ n−1or else dim(H(R)/K H ) = n and there is a homomorphism from H(R) ontoPSL(2,R) [Prasad 1976, Theorem B].

Suppose that dim(H(R)/K H ) ≤ n − 1. We know that 0 acts properly onH(R)/K H and that0 contains Zn−1. The quotient Zn−1

\H(R)/K H is a manifoldof dimension n− 1 with the homotopy type of the (n− 1)-torus (in particular,with nonzero cohomology in dimension n− 1), and so it must be compact. SoZn−1 has finite index in 0, contradicting our assumption.

So dim(H(R)/K H )= n and H(R) maps onto PSL(2,R). We can assume thatHQ is simply connected. Since H is Q-simple, H is equal to the restriction ofscalars RK/QL for some number field K and some absolutely simple and simplyconnected group L over K [Tits 1966, Section 3.1]. Since H(R) maps ontoPSL(2,R), L must be a form of SL(2). We showed that G ∼= 0 has virtual coho-mological dimension at most n−1, and so 0 must be a noncocompact subgroup ofH(R). Equivalently, H has Q-rank greater than zero [Borel and Harish-Chandra1962, Lemma 11.4, Theorem 11.6], and so rankK (L)= rankQ(H) is greater thanzero. Therefore, L is isomorphic to SL(2) over K .

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It follows that 0 is commensurable with SL(2, oK ), where oK is the ring ofintegers of K . So we can assume that 0 contains the semidirect product

o∗K n oK =

{(a b0 1/a

)}⊂ SL(2, oK ).

If the group of units o∗K has positive rank, then o∗K noK is a solvable group whichis not virtually abelian. So the group of units is finite, which means that K is eitherQ or an imaginary quadratic field, by Dirichlet. If K is imaginary quadratic, thenHQ = RK/Q SL(2) and H(R)= SL(2,C), which does not map onto PSL(2,R).Therefore K = Q and HQ = SL(2). It follows that 0 is commensurable withSL(2,Z). So 0 is commensurable with a free group. This contradicts that G ∼=0contains Zn−1 with n ≥ 3. �

Proof of Corollary 7.2. Let M be the Picard lattice of X , that is, M = Pic(X)with the intersection form. Then M has signature (1, n) by the Hodge indextheorem, where n ≥ 3 since X has Picard number at least 4.

For an elliptic fibration X→ P1 with no reducible fibers, the Mordell–Weilgroup of the fibration has rank ρ(X)− 2 = n− 1 by the Shioda–Tate formula[Shioda 1972, Corollary 1.5], which is easy to check in this case. So the firstelliptic fibration of X gives an inclusion of Zn−1 into G =Aut∗(X). The secondelliptic fibration gives an inclusion of Za into G for some a > 0. In the action ofG on hyperbolic n-space, the Mordell–Weil group of each elliptic fibration is agroup of parabolic transformations fixing the point at infinity that corresponds tothe class e∈M of a fiber (which has 〈e, e〉= 0). Since a parabolic transformationfixes only one point of the sphere at infinity, the subgroups Zn−1 and Za in Gintersect only in the identity. It follows that the subgroup Zn−1 has infinite indexin G.

We are given that X contains a (−2)-curve C . I claim that C has infinitelymany translates under the Mordell–Weil group Zn−1. Indeed, any curve withfinitely many orbits under Zn−1 must be contained in a fiber of X→ P1. Sinceall fibers are irreducible, the fibers have self-intersection 0, not −2. Thus Xcontains infinitely many (−2)-curves. Therefore the group

W ⊂ O(M)

generated by reflections in (−2)-vectors is infinite. Here W acts simply transi-tively on the Weyl chambers of the positive cone (separated by hyperplanes v⊥

with v a (−2)-vector), whereas G = Aut∗(X) preserves one Weyl chamber, theample cone of X . So G and W intersect only in the identity. Since W is infinite,G has infinite index in O(M). By Theorem 7.1, G is not commensurable withan arithmetic group. �


Proof of Example 7.3. The given curve C in the linear system |O(4, 4)| =|− 2KP1×P1 | is smooth. One can check this with Macaulay 2, for example.Therefore the double cover X of P1

× P1 ramified along C is a smooth K3surface. The two projections from X to P1 are elliptic fibrations. Typically,such a double cover π : X → P1

× P1 would have Picard number 2, but thecurve C has been chosen to be tangent at 4 points to each of two curves ofdegree (1, 1), D1 = {xv = yu} and D2 = {xv = −yu}. (These points are[x, y] = [u, v] equal to [1, 1], [1, 2], [1,−1], [1,−2] on D1 and [x, y] = [u,−v]equal to [1, 1], [1, 2], [1,−1], [1,−2] on D2.) It follows that the double coveringis trivial over D1 and D2, outside the ramification curve C : the inverse imagein X of each curve Di is a union of two curves, π−1(Di ) = Ei ∪ Fi , meetingtransversely at 4 points. The smooth rational curves E1, F1, E2, F2 on X are(−2)-curves, since X is a K3 surface.

The curves D1 and D2 meet transversely at the two points [x, y] = [u, v]equal to [1, 0] or [0, 1]. Let us compute that the double covering

π : X→ P1×P1

is trivial over the union of D1 and D2 (outside the ramification curve C). Indeed,if we write X as w2

= f (x, y, z, w) where f is the given polynomial of degree(4, 4), then a section of π over D1 ∪ D2 is given by

w = 4x2u2− 5x2v2

+ y2v2.

We can name the curves Ei , Fi so that the image of this section is E1∪E2 and theimage of the sectionw=−(4x2u2

−5x2v2+y2v2) is F1∪F2. Then E1 and F2 are

disjoint. So the intersection form among the divisors π∗O(1,0),π∗O(0,1),E1,F2

on X is given by 0 2 1 12 0 1 11 1 −2 01 1 0 −2

Since this matrix has determinant −32, not zero, X has Picard number at least 4.

Finally, we compute that the two projections from C ⊂P1×P1 to P1 are each

ramified over 24 distinct points in P1. It follows that all fibers of our two ellipticfibrations X→ P1 are irreducible. By Corollary 7.2, the automorphism groupof X (over C, or equivalently over Q) is not commensurable with an arithmeticgroup. Our calculations have all worked over Q, and so Corollary 7.2 also givesthat Aut(XQ) is not commensurable with an arithmetic group. �

418 BURT TOTARO

8. Klt pairs

We will see that the previous results can be generalized from Calabi–Yau varietiesto a broader class of varieties using the language of pairs. For the rest of thepaper, we work over the complex numbers.

A normal variety X is Q-factorial if for every point p and every codimension-one subvariety S through p, there is a regular function on some neighborhood ofp that vanishes exactly on S (to some positive order).

Definition 8.1. A pair (X,1) is a Q-factorial projective variety X with aneffective R-divisor 1 on X .

Notice that1 is an actual R-divisor1=∑

ai1i , not a numerical equivalenceclass of divisors. We think of K X +1 as the canonical bundle of the pair (X,1).The following definition picks out an important class of “mildly singular” pairs.

Definition 8.2. A pair (X,1) is klt (Kawamata log terminal) if the followingholds. Let π : X→ X be a resolution of singularities. Suppose that the union ofthe exceptional set of π (the subset of X where π is not an isomorphism) withπ−1(1) is a divisor with simple normal crossings. Define a divisor 1 on X by

K X + 1= π∗(K X +1).

We say that (X,1) is klt if all coefficients of 1 are less than 1. This property isindependent of the choice of resolution.

Example 8.3. A surface X = (X, 0) is klt if and only if X has only quotientsingularities [Kollár and Mori 1998, Proposition 4.18].

Example 8.4. For a smooth variety X and 1 a divisor with simple normalcrossings (and some coefficients), the pair (X,1) is klt if and only if 1 hascoefficients less than 1.

All the main results of minimal model theory, such as the cone theorem,generalize from smooth varieties to klt pairs. For example, the Fano case of thecone theorem becomes [Kollár and Mori 1998, Theorem 3.7]:

Theorem 8.5. Let (X,1) be a klt Fano pair, meaning that −(K X +1) is ample.Then Curv(X) (and hence the dual cone Nef(X)) is rational polyhedral.

Notice that the conclusion does not involve the divisor 1. This shows thepower of the language of pairs. A variety X may not be Fano, but if we canfind an R-divisor 1 that makes (X,1) a klt Fano pair, then we get the sameconclusion (that the cone of curves and the nef cone are rational polyhedral) asif X were Fano.


Example 8.6. Let X be the blow-up of P2 at any number of points on a smoothconic. As an exercise, the reader can write down an R-divisor1 such that (X,1)is a klt Fano pair. This proves that the nef cone of X is rational polyhedral, asproved by other methods in [Galindo and Monserrat 2005, Corollary 3.3; Mukai2005; Castravet and Tevelev 2006]. These surfaces are definitely not Fano if weblow up 6 or more points. Their Betti numbers are unbounded, in contrast to thesmooth Fano surfaces.

More generally, Testa, Várilly-Alvarado, and Velasco proved that every smoothprojective rational surface X with−K X big has finitely generated Cox ring [Testaet al. 2009]. Finite generation of the Cox ring (the ring of all sections of all linebundles) is stronger than the nef cone being rational polyhedral, by the analysisof [Hu and Keel 2000]. Chenyang Xu showed that a rational surface with −K X

big need not have any divisor 1 with (X,1) a klt Fano pair [Testa et al. 2009].I do not know whether the blow-ups of higher-dimensional projective spacesconsidered by in [Mukai 2005] and [Castravet and Tevelev 2006] have a divisor1 with (X,1) a klt Fano pair.

It is therefore natural to extend the Morrison–Kawamata cone conjecture fromCalabi–Yau varieties to Calabi–Yau pairs (X,1), meaning that K X +1 ≡ 0.The conjecture is reasonable, since we can prove it in dimension 2 [Totaro 2010].

Theorem 8.7. Let (X,1) be a klt Calabi–Yau pair of dimension 2. ThenAut(X,1) (and also Aut(X)) acts with a rational polyhedral fundamental do-main on the cone Nef(X)⊂ N 1(X).

Here is a more concrete consequence of Theorem 8.7:

Corollary 8.8 [Totaro 2010]. Let (X,1) be a klt Calabi–Yau pair of dimension2. Then there are only finitely many contractions of X up to automorphisms of X.Related to that: Aut(X) has finitely many orbits on the set of curves in X withnegative self-intersection.

This was shown in one class of examples [Dolgachev and Zhang 2001]. Theseresults are false for surfaces in general, even for some smooth rational surfaces:

Example 8.9. Let X be the blow-up of P2 at 9 very general points. Then Nef(X)is not rational polyhedral, since X contains infinitely many (−1)-curves. ButAut(X)= 1 [Gizatullin 1980, Proposition 8], and so the conclusion fails for X .

Moreover, let 1 be the unique smooth cubic curve in P2 through the 9 points,with coefficient 1. Then −K X ≡1, and so (X,1) is a log-canonical (and evencanonical) Calabi–Yau pair. The theorem therefore fails for such pairs.

We now give a classical example (besides the case 1 = 0 of Calabi–Yausurfaces) where Theorem 8.7 applies.

420 BURT TOTARO

Example 8.10. Let X be the blow-up of P2 at 9 points p1, . . . , p9 which arethe intersection of two cubic curves. Then taking linear combinations of the twocubics gives a P1-family of elliptic curves through the 9 points. These curvesbecome disjoint on the blow-up X , and so we have an elliptic fibration X→ P1.This morphism is given by the linear system |− K X |. Using that, we see that the(−1)-curves on X are exactly the sections of the elliptic fibration X→ P1.

In most cases, the Mordell–Weil group of X → P1 is .= Z8. So X containsinfinitely many (−1)-curves, and so the cone Nef(X) is not rational polyhedral.But Aut(X) acts transitively on the set of (−1)-curves, by translations using thegroup structure on the fibers of X→ P1. That leads to the proof of Theorem 8.7in this example. (The theorem applies, in the sense that there is an R-divisor 1with (X,1) klt Calabi–Yau: let 1 be the sum of two smooth fibers of X→ P1

with coefficients 12 , for example.)

9. The cone conjecture in dimension greater than 2

In higher dimensions, the cone conjecture also predicts that a klt Calabi–Yaupair (X,1) has only finitely many small Q-factorial modifications X 99K X1 upto pseudo-automorphisms of X . (See [Kawamata 1997; Totaro 2010] for the fullstatement of the cone conjecture in higher dimensions.) A pseudo-automorphismmeans a birational automorphism which is an isomorphism in codimension 1.

More generally, the conjecture implies that X has only finitely many birationalcontractions X 99K Y modulo pseudo-automorphisms of X , where a birationalcontraction means a dominant rational map that extracts no divisors. There canbe infinitely many small modifications if we do not divide out by the groupPsAut(X) of pseudo-automorphisms of X .

Kawamata proved a relative version of the cone conjecture for a 3-fold X witha K3 fibration or elliptic fibration X→ S [Kawamata 1997]. Here X can haveinfinitely many minimal models (or small modifications) over S, but it has onlyfinitely many modulo PsAut(X/S).

This is related to other finiteness problems in minimal model theory. Weknow that a klt pair (X,1) has only finitely many minimal models if 1 is big[Birkar et al. 2010, Corollary 1.1.5]. It follows that a variety of general typehas a finite set of minimal models. A variety of nonmaximal Kodaira dimensioncan have infinitely many minimal models [Reid 1983, Section 6.8; 1997]. Butit is conjectured that every variety X has only finitely many minimal modelsup to isomorphism, meaning that we ignore the birational identification withX . Kawamata’s results on Calabi–Yau fiber spaces imply at least that 3-foldsof positive Kodaira dimension have only finitely many minimal models up toisomorphism [Kawamata 1997, Theorem 4.5]. If the abundance conjecture


[Kollár and Mori 1998, Corollary 3.12] holds (as it does in dimension 3), thenevery nonuniruled variety has an Iitaka fibration where the fibers are Calabi–Yau. The cone conjecture for Calabi–Yau fiber spaces (plus abundance) impliesfiniteness of minimal models up to isomorphism for arbitrary varieties.

The cone conjecture is wide open for Calabi–Yau 3-folds, despite significantresults by Oguiso and Peternell [1998], Szendroi [1999], Uehara [2004], andWilson [1994]. Hassett and Tschinkel recently [2010] checked the conjecturefor a class of holomorphic symplectic 4-folds.

10. Outline of the proof of Theorem 8.7

The proof of Theorem 8.7 gives a good picture of the Calabi–Yau pairs ofdimension 2. We summarize the proof from [Totaro 2010]. In most cases, if(X,1) is a Calabi–Yau pair, then X turns out to be rational. It is striking that themost interesting case of the theorem is proved by reducing properties of certainrational surfaces to the Torelli theorem for K3 surfaces.

Let (X,1) be a klt Calabi–Yau pair of dimension 2. That is, K X +1≡ 0, orequivalently

−K X ≡1,

where 1 is effective. We can reduce to the case where X is smooth by taking asuitable resolution of (X,1).

If 1= 0, then X is a smooth Calabi–Yau surface, and the result is known bySterk, using the Torelli theorem for K3 surfaces. So assume that 1 6= 0. Then Xhas Kodaira dimension

κ(X) := κ(X, K X )

equal to −∞. With one easy exception, Nikulin showed that our assumptionsimply that X is rational [Alexeev and Mori 2004, Lemma 1.4]. So assume thatX is rational from now on.

We have three main cases for the proof, depending on whether the Iitakadimension κ(X,−K X ) is 0, 1, or 2. (It is nonnegative because −K X ∼R 1 ≥

0.) By definition, the Iitaka dimension κ(X, L) of a line bundle L is −∞ ifh0(X,mL) = 0 for all positive integers m. Otherwise, κ(X, L) is the naturalnumber r such that there are positive integers a, b and a positive integer m0 withamr≤ h0(X,mL) ≤ bmr for all positive multiples m of m0 [Lazarsfeld 2004,

Corollary 2.1.38].

10.1. Case where κ(X,−KX)= 2. That is,−K X is big. In this case, there is anR-divisor 0 such that (X, 0) is klt Fano. Therefore Nef(X) is rational polyhedralby the cone theorem, and hence the group Aut∗(X) is finite. So Theorem 8.7 istrue in a simple way. More generally, for (X, 0) klt Fano of any dimension, the

422 BURT TOTARO

Cox ring of X is finitely generated, by Birkar, Cascini, Hacon, and McKernan[2010].

This proof illustrates an interesting aspect of working with pairs: rather thanFano being a different case from Calabi–Yau, Fano becomes a special case ofCalabi–Yau. That is, if (X, 0) is a klt Fano pair, then there is another effectiveR-divisor 1 with (X,1) a klt Calabi–Yau pair.

10.2. Case where κ(X,−KX)=1. In this case, some positive multiple of−K X

gives an elliptic fibration X→P1, not necessarily minimal. Here Aut∗(X) equalsthe Mordell–Weil group of X→ P1 up to finite index, and so Aut∗(X) .= Zn forsome n. This generalizes the example of P2 blown up at the intersection of twocubic curves.

The (−1)-curves in X are multisections of X→ P1 of a certain fixed degree.One shows that Aut(X) has only finitely many orbits on the set of (−1)-curvesin X . This leads to the statement of Theorem 8.7 in terms of cones.

10.3. Case where κ(X,−KX)= 0. This is the hardest case. Here Aut∗(X) canbe a fairly general group acting on hyperbolic space; in particular, it can behighly nonabelian.

Here −K X ≡1 where the intersection pairing on the curves in 1 is negativedefinite. We can contract all the curves in 1, yielding a singular surface Y with−KY ≡ 0. Note that Y is klt and hence has quotient singularities, but it musthave worse than ADE singularities, because it is a singular Calabi–Yau surfacethat is rational.

Let I be the “global index” of Y , the least positive integer with I KY Cartierand linearly equivalent to zero. Then

Y = M/(Z/I )

for some Calabi–Yau surface M with ADE singularities. The minimal resolutionof M is a smooth Calabi–Yau surface. Using the Torelli theorem for K3 surfaces,this leads to the proof of the theorem for M and then for Y , by Oguiso andSakurai [2001, Corollary 1.9].

Finally, we have to go from Y to its resolution of singularities, the smoothrational surface X . Here Nef(X) is more complex than Nef(Y ): X typicallycontains infinitely many (−1)-curves, whereas Y has none (because KY ≡ 0).Nonetheless, since we know “how big” Aut(Y ) is (up to finite index), we canshow that the group

Aut(X,1)= Aut(Y )

has finitely many orbits on the set of (−1)-curves. This leads to the proof ofTheorem 8.7 for (X,1). QED


11. Example

Here is an example of a smooth rational surface with a highly nonabelian (discrete)automorphism group, considered by Zhang [1991, Theorem 4.1, p. 438], Blache[1995, Theorem C(b)(2)], and [2010, Section 2]. This is an example of the lastcase in the proof of Theorem 8.7, where κ(X,−K X ) = 0. We will also see asingular rational surface whose nef cone is round, of dimension 4.

Let X be the blow-up of P2 at the 12 points: [1, ζ i , ζ j] for i, j ∈Z/3, [1, 0, 0],

[0, 1, 0], [0, 0, 1]. Here ζ is a cube root of 1. (This is the dual of the “Hesseconfiguration” [Dolgachev 2004, Section 4.6]. There are 9 lines L1, . . . , L9

through quadruples of the 12 points in P2.)On P2, we have

−KP2 ≡ 3H ≡9∑

i=1

13 L i .

On the blow-up X , we have

−K X ≡

9∑i=1

13 L i ,

where L1, . . . , L9 are the proper transforms of the 9 lines, which are now disjointand have self-intersection number −3. Thus

(X,∑9

i=113 L i

)is a klt Calabi–Yau

pair.Section 10.3 shows how to analyze X : contract the 9 (−3)-curves L i on

X . This gives a rational surface Y with 9 singular points (of type 13(1, 1)) and

ρ(Y )= 4. We have −KY ≡ 0, so Y is a klt Calabi–Yau surface which is rational.We have 3KY ∼ 0, and so Y ∼= M/(Z/3) with M a Calabi–Yau surface withADE singularities. It turns out that M is smooth, M ∼= E × E where E is theFermat cubic curve

E ∼= C/Z[ζ ] ∼= {[x, y, z] ∈ P2: x3+ y3= z3},

and Z/3 acts on E × E as multiplication by (ζ, ζ ) [Totaro 2010, Section 2].Since E has endomorphism ring Z[ζ ], the group GL(2,Z[ζ ]) acts on the

abelian surface M = E × E . This passes to an action on the quotient varietyY = M/(Z/3) and hence on its minimal resolution X (which is the blow-up ofP2 at 12 points we started with). Thus the infinite, highly nonabelian discretegroup GL(2,Z[ζ ]) acts on the smooth rational surface X . This is the wholeautomorphism group of X up to finite groups (loc. cit.).

Here Nef(Y ) = Nef(M) is a round cone in R4, and so Theorem 8.7 saysthat PGL(2,Z[ζ ]) acts with finite covolume on hyperbolic 3-space. In fact, thequotient of hyperbolic 3-space by an index-24 subgroup of PGL(2,Z[ζ ]) is

424 BURT TOTARO

familiar to topologists as the complement of the figure-eight knot [Maclachlanand Reid 2003, 1.4.3, 4.7.1].

References

[Alekseevskij et al. 1993] D. V. Alekseevskij, È. B. Vinberg, and A. S. Solodovnikov, “Geometryof spaces of constant curvature”, pp. 1–138 in Geometry, vol. II, Encyclopaedia Math. Sci. 29,Springer, Berlin, 1993.

[Alexeev and Mori 2004] V. Alexeev and S. Mori, “Bounding singular surfaces of general type”,pp. 143–174 in Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000),Springer, Berlin, 2004.

[Birkar et al. 2010] C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, “Existence of minimalmodels for varieties of log general type”, J. Amer. Math. Soc. 23:2 (2010), 405–468.

[Blache 1995] R. Blache, “The structure of l.c. surfaces of Kodaira dimension zero. I”, J. AlgebraicGeom. 4:1 (1995), 137–179.

[Borcherds 1998] R. E. Borcherds, “Coxeter groups, Lorentzian lattices, and K 3 surfaces”,Internat. Math. Res. Notices 1998:19 (1998), 1011–1031.

[Borel and Harish-Chandra 1962] A. Borel and Harish-Chandra, “Arithmetic subgroups of alge-braic groups”, Ann. of Math. (2) 75 (1962), 485–535.

[Borel and Serre 1964] A. Borel and J.-P. Serre, “Théorèmes de finitude en cohomologie galoisi-enne”, Comment. Math. Helv. 39 (1964), 111–164.

[Bridson and Haefliger 1999] M. R. Bridson and A. Haefliger, Metric spaces of non-positivecurvature, Grundlehren der Mathematischen Wissenschaften 319, Springer, Berlin, 1999.

[Brown 1982] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer,New York, 1982.

[Castravet and Tevelev 2006] A.-M. Castravet and J. Tevelev, “Hilbert’s 14th problem and Coxrings”, Compos. Math. 142:6 (2006), 1479–1498.

[Debarre 2001] O. Debarre, Higher-dimensional algebraic geometry, Springer, New York, 2001.

[Dolgachev 2004] I. V. Dolgachev, “Abstract configurations in algebraic geometry”, pp. 423–462in The Fano Conference, Univ. Torino, Turin, 2004.

[Dolgachev 2008] I. V. Dolgachev, “Reflection groups in algebraic geometry”, Bull. Amer. Math.Soc. (N.S.) 45:1 (2008), 1–60.

[Dolgachev and Zhang 2001] I. V. Dolgachev and D.-Q. Zhang, “Coble rational surfaces”, Amer.J. Math. 123:1 (2001), 79–114.

[de Fernex 2010] T. de Fernex, “The Mori cone of blow-ups of the plane”, preprint, 2010.arXiv 1001.5243

[Galindo and Monserrat 2005] C. Galindo and F. Monserrat, “The total coordinate ring of asmooth projective surface”, J. Algebra 284:1 (2005), 91–101.

[Gizatullin 1980] M. H. Gizatullin, “Rational G-surfaces”, Izv. Akad. Nauk SSSR Ser. Mat. 44:1(1980), 110–144. In Russian; translated in Math. USSR Izv. 16 (1981), 103–134.

[Hartshorne 1977] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52,Springer, New York, 1977.

[Hassett and Tschinkel 2010] B. Hassett and Y. Tschinkel, “Flops on holomorphic symplecticfourfolds and determinantal cubic hypersurfaces”, J. Inst. Math. Jussieu 9:1 (2010), 125–153.


[Hu and Keel 2000] Y. Hu and S. Keel, “Mori dream spaces and GIT”, Michigan Math. J. 48(2000), 331–348.

[Iskovkikh and Prokhorov 1999] V. Iskovkikh and Y. Prokhorov, “Fano varieties”, pp. 1–247 inAlgebraic geometry, vol. V, Encyclopaedia of Mathematical Sciences 47, Springer, Berlin, 1999.

[Kawamata 1997] Y. Kawamata, “On the cone of divisors of Calabi–Yau fiber spaces”, Internat. J.Math. 8:5 (1997), 665–687.

[Kollár and Mori 1998] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cam-bridge Tracts in Mathematics 134, Cambridge University Press, Cambridge, 1998.

[Kollár et al. 1992] J. Kollár, Y. Miyaoka, and S. Mori, “Rational connectedness and boundednessof Fano manifolds”, J. Differential Geom. 36:3 (1992), 765–779.

[Kovács 1994] S. J. Kovács, “The cone of curves of a K 3 surface”, Math. Ann. 300:4 (1994),681–691.

[Lazarsfeld 2004] R. Lazarsfeld, Positivity in algebraic geometry, I: classical setting: line bundlesand linear series, Ergebnisse der Math. (3) 48, Springer, Berlin, 2004.

[Maclachlan and Reid 2003] C. Maclachlan and A. W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics 219, Springer, New York, 2003.

[Mazur 1993] B. Mazur, “On the passage from local to global in number theory”, Bull. Amer.Math. Soc. (N.S.) 29:1 (1993), 14–50.

[Mukai 2005] S. Mukai, “Finite generation of the Nagata invariant rings in A-D-E cases”, preprint1502, RIMS, 2005.

[Mumford 1970] D. Mumford, Abelian varieties, Tata Studies in Mathematics 5, Tata Institute ofFundamental Research, Bombay, 1970.

[Namikawa 1985] Y. Namikawa, “Periods of Enriques surfaces”, Math. Ann. 270:2 (1985), 201–222.

[Nikulin 1979] V. V. Nikulin, “Integer symmetric bilinear forms and some of their geometricapplications”, Izv. Akad. Nauk SSSR Ser. Mat. 43:1 (1979), 111–177. In Russian; translated inMath. USSR Izv. 14 (1979), 103–167.

[Oguiso and Peternell 1998] K. Oguiso and T. Peternell, “Calabi–Yau threefolds with positivesecond Chern class”, Comm. Anal. Geom. 6:1 (1998), 153–172.

[Oguiso and Sakurai 2001] K. Oguiso and J. Sakurai, “Calabi–Yau threefolds of quotient type”,Asian J. Math. 5:1 (2001), 43–77.

[Prasad 1976] G. Prasad, “Discrete subgroups isomorphic to lattices in Lie groups”, Amer. J. Math.98:4 (1976), 853–863.

[Reid 1983] M. Reid, “Minimal models of canonical 3-folds”, pp. 131–180 in Algebraic varietiesand analytic varieties (Tokyo, 1981), Adv. Stud. Pure Math. 1, North-Holland, Amsterdam, 1983.

[Serre 1979] J.-P. Serre, “Arithmetic groups”, pp. 105–136 in Homological group theory (Durham,1977), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press, Cambridge, 1979.Reprinted in his Œuvres, v. 3, Springer (1986), 503–534.

[Shioda 1972] T. Shioda, “On elliptic modular surfaces”, J. Math. Soc. Japan 24 (1972), 20–59.

[Sterk 1985] H. Sterk, “Finiteness results for algebraic K 3 surfaces”, Math. Z. 189:4 (1985),507–513.

[Szendroi 1999] B. Szendroi, “Some finiteness results for Calabi–Yau threefolds”, J. LondonMath. Soc. (2) 60:3 (1999), 689–699.

[Testa et al. 2009] D. Testa, A. Várilly-Alvarado, and M. Velasco, “Big rational surfaces”, preprint,2009. To appear in Math. Ann. arXiv 0901.1094

426 BURT TOTARO

[Tits 1966] J. Tits, “Classification of algebraic semisimple groups”, pp. 33–62 in Algebraic groupsand discontinuous subgroups (Boulder, CO, 1965), Amer. Math. Soc., Providence, RI, 1966,1966.

[Totaro 2010] B. Totaro, “The cone conjecture for Calabi–Yau pairs in dimension 2”, Duke Math.J. 154:2 (2010), 241–263.

[Uehara 2004] H. Uehara, “Calabi–Yau threefolds with infinitely many divisorial contractions”, J.Math. Kyoto Univ. 44:1 (2004), 99–118.

[Wilson 1994] P. M. H. Wilson, “Minimal models of Calabi–Yau threefolds”, pp. 403–410 inClassification of algebraic varieties (L’Aquila, 1992), Contemp. Math. 162, Amer. Math. Soc.,Providence, RI, 1994.

[Yau 1978] S. T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complexMonge–Ampère equation, I”, Comm. Pure Appl. Math. 31:3 (1978), 339–411.

[Zhang 1991] D.-Q. Zhang, “Logarithmic Enriques surfaces”, J. Math. Kyoto Univ. 31:2 (1991),419–466.

[email protected] Department of Pure Mathematics and Mathematical

Statistics, Cambridge University, Wilberforce Road,

Cambridge CB3 0WB, England

algebraic surfaces and hyperbolic geometry

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